This application claims benefit of Japanese Application Nos. 2000-251869 and 2001-7199 filed in Japan on Aug. 23, 2000 and Jan. 16, 2001, respectively, the contents of which are incorporated by this reference.
1. Field of the Invention
The present invention relates to a variable focal length optical element and an optical system using the same. More particularly, the present invention relates to a variable focal length optical element for use in an image pickup optical system having a zooming mechanism and a focusing mechanism and also relates to an optical system using such a variable focal length optical element.
2. Discussion of Related Art
Variable-magnification image pickup optical systems using prisms with free-form surfaces are known from Japanese Patent Application Unexamined Publication Numbers [hereinafter referred to as xe2x80x9cJP(A)xe2x80x9d] 8-292372, 11-317894, 11-317895, etc.
However, JP(A) 8-292372 is arranged to perform zooming by moving a plurality of prisms. Therefore, it is necessary to ensure a space for moving the prisms in the apparatus in advance, which leads to an increase in the size of the apparatus. Further, because the prism moving mechanism needs to move the prisms rectilinearly with high accuracy, the structure becomes complicated. This causes costs to increase and hinders assembling considerably.
Similarly, JP(A) 11-317894 and 11-317895 have a zooming mechanism in which a transmission lens optical element is moved, and hence suffer from the problem that the structure becomes complicated as in the case of the above.
Thus, the above-described conventional techniques need to ensure a space for moving an optical element in advance. That is, it is necessary to ensure a useless space. Therefore, it is impossible with the conventional techniques to construct a compact variable-magnification optical system. In addition, the optical element per se produces aberrations. Therefore, it cannot be used to change optical parameters to a considerable extent.
Common zooming and focus adjustment are generally effected by moving some optical elements in the optical system along the optical axis direction. However, this method has the disadvantage that it is difficult to move the optical elements completely parallel to the optical axis. Tilt or decentration of the optical elements causes the image quality to be degraded. In particular, it has heretofore been impossible to construct a variable-magnification optical system for use with a small-sized image pickup device because the requirement for the accuracy of movement is very strict.
The present invention was made in view of the above-described problems with the prior art.
An object of the present invention is to provide an image-forming optical system or the like capable of focal length adjustment, focus adjustment, etc. with a simple arrangement in which a desired portion of an optical element is selectively utilized by moving an aperture in a direction approximately perpendicular to the optical axis.
To attain the above-described object, the present invention provides a variable focal length optical element having at least one continuous rotationally asymmetric surface. The optical power of the optical element is variable by selecting an optical path through an aperture.
In this case, it is possible to obtain an image-forming optical system capable of adjusting the focal length, the image-formation position, etc. with a simple arrangement in which a desired portion of the optical element is selectively utilized by moving the aperture in a direction approximately perpendicular to the optical axis.
It is desirable that the relationship A between the size D of the aperture and the amount of movement d of the aperture, i.e. A=D/d, should satisfy the following condition:
0.01 less than A less than 1xe2x80x83xe2x80x83(1) 
The reasons for adopting the above-described arrangements in the present invention, together with the functions thereof, will be described below.
FIG. 1 is a diagram schematically showing light rays in the meridional section of an image-forming optical system (optical element) S substantially corrected for spherical aberration, curvature of field, astigmatism, distortion, axial chromatic aberration and lateral chromatic aberration but allowed to generate large comatic aberration. Because of the large comatic aberration, light rays {circle around (1)}, {circle around (2)} and {circle around (3)} emanating from an object O at different field angles are incident on different positions on an image plane I. Accordingly, if the position of the aperture of the image-forming optical system S is moved from P1 through P2 to P3 in a direction approximately perpendicular to the optical axis (herein assumed to be an axial principal ray passing through the center of each of the apertures P1, P2 and P3 and reaching the center of the image plane, although the optical axis is defined by an axial principal ray passing through the center of the entrance pupil and reaching the center of the image plane), light rays passing through the image-forming optical system S change their passing position and angle, i.e. {circle around (1)}xe2x86x92{circle around (2)}xe2x86x92{circle around (3)}, and are incident on the image plane I at different heights H1, H2 and H3. Therefore, the image-formation magnification of the object O on the image plane I varies according to the position of the aperture. Accordingly, the magnification can be varied by moving a stop constituting each aperture in a direction approximately perpendicular to the optical axis. On the same principle, the image-formation position and the principal point position can be adjusted by moving the pupil position in a direction approximately perpendicular to the optical axis. It should be noted that the aperture position can also be moved in a direction perpendicular to the plane of the figure. Therefore, it is possible to effect zooming, for example, by movement in the plane of the figure and to perform focusing, for example, by movement in a direction perpendicular to the plane of the figure.
It is preferable that the relationship A between the size D of the aperture, which limits the cross-sectional area of a beam of light, and the amount of movement d of the aperture, i.e. A=D/d, should satisfy the following condition:
0.01 less than A less than xe2x80x83xe2x80x83(1) 
It should be noted that when the aperture is projected within the variable focal length optical element by an optical path selecting device for selecting an optical path, D and d correspond to the size of the aperture and the amount of movement thereof within the variable focal length optical element.
If A is not larger than the lower limit of the condition (1), i.e. 0.01, the amount of movement of the aperture becomes excessively large in comparison to the size of the aperture. Consequently, it becomes impossible to construct a compact optical element, which is one object to be attained by the variable focal length optical element according to the present invention. If A is not smaller than the upper limit of the condition (1), i.e. 1, the required amount of movement of the aperture cannot satisfactorily be ensured. Consequently, it becomes impossible to make the focal length variable.
It should be noted that in Example 4 (described later), D, d and A are as follows:
In addition, the present invention provides an optical system having at least two optical elements. The optical system includes a first unit and a second unit. The first unit is positioned at the object side of the optical system to form a first image. The second unit projects the first image onto an image plane. An optical path selecting device for selecting an optical path is positioned in the vicinity of the first image. Either the first or second unit includes a variable focal length optical element having at least two continuous rotationally asymmetric surfaces.
In this case, it is desirable that an optical path in the variable focal length optical element should be selected by a combination of the aperture and the optical path selecting device positioned in the vicinity of the first image.
The optical path selecting device may be a light-deflecting device positioned in the vicinity of the first image to change the position and direction of a light beam entering the second unit behind the first image, thereby adjusting at least one of the focal length, the image-formation position, the image-formation magnification, and the principal point position.
The following is a description of the reasons for adopting the above-described arrangements in the present invention, together with the functions thereof. As will be clear from FIG. 1, the arrangement in which the aperture is moved within the effective diameter of the image-forming optical system S from P1 through P2 to P3 causes the effective diameter of the image-forming optical system S to increase unfavorably. To avoid this problem, as shown in FIG. 2, an optical system (first unit) Sxe2x80x2 is added, and a deflecting mirror M is positioned in the vicinity of an intermediate image (first image) plane of the object O. By changing the deflection angle with the deflecting mirror M, the same effect as the movement of the aperture in FIG. 1 is obtained. Thus, light rays {circle around (1)}, {circle around (2)} and {circle around (3)} are incident on the image plane I at different heights H1, H2 and H3 according to the deflection angle of the deflecting mirror M to form images of the object O at different image-formation magnifications in the same way as in FIG. 1. In this case, the image-formation position and the principal point position can also be adjusted by changing the deflection angle of the deflecting mirror M. If two axes of rotation are set for the deflecting mirror M in a direction perpendicular to the plane of the figure and in the plane of the figure, for example, zooming and focusing can be effected separately.
The optical system may also be arranged as follows. The first unit is formed as a variable focal length optical element, and an aperture for forming an exit pupil is positioned in the second unit. The position of an entrance pupil conjugate to the aperture positioned in the second unit is moved relative to the first unit by the optical path selecting device positioned in the vicinity of the first image, thereby adjusting at least one of the focal length, the image-formation position, the image-formation magnification, and the principal point position.
Further, the optical path selecting device can be implemented by moving the aperture (stop) itself. It is also possible to make both the first and second units operate as a variable focal length optical element.
In addition, the present invention provides another variable focal length optical element having at least one discontinuous rotationally asymmetric surface. The optical power of the optical element is variable by selecting an optical path through an aperture.
In this case, it is possible to obtain an image-forming optical system capable of adjusting the focal length, the image-formation position, etc. with a simple arrangement in which a desired portion of the optical element is selectively utilized by moving the aperture in a direction approximately perpendicular to the optical axis.
In the foregoing optical system and another variable focal length optical element according to the present invention, if an optical path selecting device is used, it is preferable that the width a of oscillation of the axial principal ray caused by the optical path selecting device should satisfy the following condition:
0xc2x0 less than xcex1 less than 90xc2x0xe2x80x83xe2x80x83(2) 
If xcex1 is not larger than the lower limit of the condition (2), i.e. 0xc2x0, it becomes impossible to select an optical path and hence impossible to change optical parameters. If xcex1 is not smaller than the upper limit, i.e. 90xc2x0, the size of the variable focal length optical element becomes unfavorably large.
It is even more desirable to satisfy the following condition:
0xc2x0 less than xcex1 less than 45xc2x0xe2x80x83xe2x80x83(2-1) 
If the foregoing optical system and another variable focal length optical element according to the present invention are used to perform zooming, it is still more desirable to satisfy the following condition:
10xc2x0 less than xcex1 less than 45xc2x0xe2x80x83xe2x80x83(3) 
If xcex1 is not larger than the lower limit of the condition (3), i.e. 10xc2x0, local power change in the continuous rotationally asymmetric surface becomes unfavorably large. Consequently, it becomes difficult to obtain a favorable aberration condition throughout the image field. If xcex1 is not smaller than the upper limit, i.e. 45xc2x0, the size of the variable focal length optical element becomes unfavorably large.
If the foregoing optical system and another variable focal length optical element according to the present invention are used to effect focusing, it is important to satisfy the following condition:
0xc2x0 less than xcex1 less than 30xc2x0xe2x80x83xe2x80x83(4) 
The upper limit of the above condition (4) is set at 30xc2x0 because focusing does not need to change the focal length of the optical system to a considerable extent. If xcex1 is not smaller than the upper limit, i.e. 30xc2x0, the size of the variable focal length optical element becomes unfavorably large.
It should be noted that the values of ax in Examples 2, 3, 5 to 11 (described later) are as follows:
It has been stated above that the variable focal length optical element or optical system according to the present invention can perform zooming and so forth from the viewpoint of aberrations. However, the present invention can also be described as being arranged as follows. The optical power in the effective surface of the optical element or the optical system is arranged to change continuously in the surface, and the working area of the optical element or the optical system is selected by movement of the aperture (pupil) in a direction perpendicular to the optical axis, thereby continuously or discontinuously adjusting optical parameters of the image-forming optical system, such as the magnification or focal length, the principal point position and so forth, without moving another optical element. It should be noted that an arrangement for discontinuously adjusting optical parameters of the optical element or the optical system may be formed by using a single free-form surface. However, it is also possible to use a discontinuous free-form surface formed by juxtaposing a plurality of free-form surfaces (a description of free-form surfaces will be given later).
It should be noted that the optical element or optical system capable of adjusting at least one of the focal length, the image-formation position, the image-formation magnification, and the principal point position by movement of the aperture position in a direction approximately perpendicular to the optical axis as stated above can be used to form, for example, a refracting optical system, a reflecting optical system or a catadioptric (reflective refracting) optical system that has at least one continuous rotationally asymmetric surface as in Examples of the present invention (described later).
In the present invention, a free-form surface is used as a typical example of a rotationally asymmetric surface. A free-form surface is defined by the following equation. The Z-axis of the defining equation is the axis of a free-form surface.                     Z        =                                            cr              2                        /                          [                              1                +                                                                            xe2x80x83                                                        ⁢                                      {                                          1                      -                                                                        (                                                      1                            +                            k                                                    )                                                ⁢                                                  c                          2                                                ⁢                                                  r                          2                                                                                      }                                                              ]                                +                                    ∑                              j                =                2                            66                        ⁢                          xe2x80x83                        ⁢                                          C                j                            ⁢                              X                m                            ⁢                              Y                n                                                                        (        a        )            
In the equation (a), the first term is a spherical surface term, and the second term is a free-form surface term.
In the spherical surface term:
c: the curvature at the vertex
k: a conic constant
r={square root over ( )}(X2+Y2)
The free-form surface term is given by                                           ∑                          j              =              2                        66                    ⁢                      xe2x80x83                    ⁢                                    C              j                        ⁢                          X              m                        ⁢                          Y              n                                      =                  xe2x80x83                ⁢                                            C              2                        ⁢            X                    +                                    C              3                        ⁢            Y                    +                                    C              4                        ⁢                          X              2                                +                                    C              5                        ⁢            XY                    +                                    C              6                        ⁢                          Y              2                                +                                                  xe2x80x83                ⁢                                            C              7                        ⁢                          X              3                                +                                    C              8                        ⁢                          X              2                        ⁢            Y                    +                                    C              9                        ⁢                          XY              2                                +                                    C              10                        ⁢                          Y              3                                +                                                  xe2x80x83                ⁢                                            C              11                        ⁢                          X              4                                +                                    C              12                        ⁢                          X              3                        ⁢            Y                    +                                    C              13                        ⁢                          X              2                        ⁢                          Y              2                                +                                    C              14                        ⁢                          XY              3                                +                                                  xe2x80x83                ⁢                                            C              15                        ⁢                          Y              4                                +                                    C              16                        ⁢                          X              5                                +                                    C              17                        ⁢                          X              4                        ⁢            Y                    +                                    C              18                        ⁢                          X              3                        ⁢                          Y              2                                +                                                  xe2x80x83                ⁢                                            C              19                        ⁢                          X              2                        ⁢                          Y              3                                +                                    C              20                        ⁢                          XY              4                                +                                    C              21                        ⁢                          Y              5                                +                                    C              22                        ⁢                          X              6                                +                                                  xe2x80x83                ⁢                                            C              23                        ⁢                          X              5                        ⁢            Y                    +                                    C              24                        ⁢                          X              4                        ⁢                          Y              2                                +                                    C              25                        ⁢                          X              3                        ⁢                          Y              3                                +                                                  xe2x80x83                ⁢                                            C              26                        ⁢                          X              2                        ⁢                          Y              4                                +                                    C              27                        ⁢                          XY              5                                +                                    C              28                        ⁢                          Y              6                                +                                                  xe2x80x83                ⁢                                            C              29                        ⁢                          X              7                                +                                    C              30                        ⁢                          X              6                        ⁢            Y                    +                                    C              31                        ⁢                          X              5                        ⁢                          Y              2                                +                                    C              32                        ⁢                          X              4                        ⁢                          Y              3                                +                                                  xe2x80x83                ⁢                                            C              33                        ⁢                          X              3                        ⁢                          Y              4                                +                                    C              34                        ⁢                          X              2                        ⁢                          Y              5                                +                                    C              35                        ⁢                          XY              6                                +                                    C              36                        ⁢                                          Y                7                            ⁢                              xe2x80x83                            .                              xe2x80x83                            .                              xe2x80x83                            .                                          
where Cj (j is an integer of 2 or higher) are coefficients.
In general, the above-described free-form surface does not have planes of symmetry in both the XZ- and YZ-planes. However, a free-form surface having only one plane of symmetry parallel to the YZ-plane is obtained by making all terms of odd-numbered degrees with respect to X zero. A free-form surface having only one plane of symmetry parallel to the XZ-plane is obtained by making all terms of odd-numbered degrees with respect to Y zero.
In addition, free-form surfaces as the above-described surfaces with a rotationally asymmetric curved surface configuration may be defined by Zernike polynomials. That is, the configuration of a free-form surface may be defined by the following equation (b). The Z-axis of the defining equation (b) is the axis of Zernike polynomial. A rotationally asymmetric surface is defined by polar coordinates of the height of the Z-axis with respect to the XY-plane. In the equation (b), R is the distance from the Z-axis in the XY-plane, and A is the azimuth angle about the Z-axis, which is expressed by the angle of rotation measured from the X-axis.
x=Rxc3x97cos(A)
y=Rxc3x97sin(A)                                                         Z              =                              xe2x80x83                            ⁢                                                D                  2                                +                                                      D                    3                                    ⁢                  R                  ⁢                                      xe2x80x83                                    ⁢                                      cos                    ⁡                                          (                      A                      )                                                                      +                                                      D                    4                                    ⁢                  R                  ⁢                                      xe2x80x83                                    ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    5                                    ⁢                                      R                    2                                    ⁢                                      cos                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                      D                    6                                    ⁡                                      (                                                                  R                        2                                            -                      1                                        )                                                  +                                                      D                    7                                    ⁢                                      R                    2                                    ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    8                                    ⁢                                      R                    3                                    ⁢                                      cos                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                                            D                      9                                        ⁡                                          (                                                                        3                          ⁢                                                      R                            3                                                                          -                                                  2                          ⁢                          R                                                                    )                                                        ⁢                                      cos                    ⁡                                          (                      A                      )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      10                                        ⁡                                          (                                                                        3                          ⁢                                                      R                            3                                                                          -                                                  2                          ⁢                          R                                                                    )                                                        ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                      D                    11                                    ⁢                                      R                    3                                    ⁢                                      sin                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    12                                    ⁢                                      R                    4                                    ⁢                                      cos                    ⁡                                          (                                              4                        ⁢                        A                                            )                                                                      +                                                                            D                      13                                        ⁡                                          (                                                                        4                          ⁢                                                      R                            4                                                                          -                                                  3                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      cos                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    14                                    ⁡                                      (                                                                  6                        ⁢                                                  R                          4                                                                    -                                              6                        ⁢                                                  R                          2                                                                    +                      1                                        )                                                  +                                                                            D                      15                                        ⁡                                          (                                                                        4                          ⁢                                                      R                            4                                                                          -                                                  3                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    16                                    ⁢                                      R                    4                                    ⁢                                      sin                    ⁡                                          (                                              4                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    17                                    ⁢                                      R                    5                                    ⁢                                      cos                    ⁡                                          (                                              5                        ⁢                        A                                            )                                                                      +                                                                            D                      18                                        ⁡                                          (                                                                        5                          ⁢                                                      R                            5                                                                          -                                                  4                          ⁢                                                      R                            3                                                                                              )                                                        ⁢                                      cos                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      19                                        ⁡                                          (                                                                        10                          ⁢                                                      R                            5                                                                          -                                                  12                          ⁢                                                      R                            3                                                                          +                                                  3                          ⁢                          R                                                                    )                                                        ⁢                                      cos                    ⁡                                          (                      A                      )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      20                                        ⁡                                          (                                                                        10                          ⁢                                                      R                            5                                                                          -                                                  12                          ⁢                                                      R                            3                                                                          +                                                  3                          ⁢                          R                                                                    )                                                        ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      21                                        ⁡                                          (                                                                        5                          ⁢                                                      R                            5                                                                          -                                                  4                          ⁢                                                      R                            3                                                                                              )                                                        ⁢                                      sin                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                      D                    22                                    ⁢                                      R                    5                                    ⁢                                      sin                    ⁡                                          (                                              5                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    23                                    ⁢                                      R                    6                                    ⁢                                      cos                    ⁡                                          (                                              6                        ⁢                        A                                            )                                                                      +                                                                            D                      24                                        ⁡                                          (                                                                        6                          ⁢                                                      R                            6                                                                          -                                                  5                          ⁢                                                      R                            4                                                                                              )                                                        ⁢                                      cos                    ⁡                                          (                                              4                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      25                                        ⁡                                          (                                                                        15                          ⁢                                                      R                            6                                                                          -                                                  20                          ⁢                                                      R                            4                                                                          +                                                  6                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      cos                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                    D                    26                                    ⁡                                      (                                                                  20                        ⁢                                                  R                          6                                                                    -                                              30                        ⁢                                                  R                          4                                                                    +                                              12                        ⁢                                                  R                          2                                                                    -                      1                                        )                                                  +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      27                                        ⁡                                          (                                                                        15                          ⁢                                                      R                            6                                                                          -                                                  20                          ⁢                                                      R                            4                                                                          +                                                  6                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                                                                        xe2x80x83                            ⁢                                                                                          D                      28                                        ⁡                                          (                                                                        6                          ⁢                                                      R                            6                                                                          -                                                  5                          ⁢                                                      R                            4                                                                                              )                                                        ⁢                                      sin                    ⁡                                          (                                              4                        ⁢                        A                                            )                                                                      +                                                      D                    29                                    ⁢                                      R                    6                                    ⁢                                                            sin                      ⁡                                              (                                                  6                          ⁢                          A                                                )                                                              ⁢                                          xe2x80x83                                        .                                          xe2x80x83                                        .                                          xe2x80x83                                        .                                                                                                          (        b        )            
where Dm (m is an integer of 2 or higher) are coefficients.
It should be noted that to design an optical system symmetric with respect to the X-axis direction, D4, D5, D6, D10, D11, D12, D13, D14, D20, D21, D22 . . . should be used.
The above defining equations are shown to exemplify surfaces with a rotationally asymmetric curved surface configuration. Therefore, the same advantageous effects can be obtained for any other defining equation that expresses such a rotationally asymmetric curved surface configuration.
It should be noted that other examples of defining equations for free-form surfaces include the following defining equation (c):
Z=xcexa3xcexa3CnmXY 
Assuming that k=7 (polynomial of degree 7), for example, a free-form surface is expressed by an expanded form of the above equation as follows:                                                         Z              =                              xe2x80x83                            ⁢                                                C                  2                                +                                                      C                    3                                    ⁢                  Y                                +                                                      C                    4                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    5                                    ⁢                                      Y                    2                                                  +                                                      C                    6                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    7                                    ⁢                                      X                    2                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    8                                    ⁢                                      Y                    3                                                  +                                                      C                    9                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    10                                    ⁢                                      YX                    2                                                  +                                                      C                    11                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    12                                    ⁢                                      Y                    4                                                  +                                                      C                    13                                    ⁢                                      Y                    3                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    14                                    ⁢                                      Y                    2                                    ⁢                                      X                    2                                                  +                                                      C                    15                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                      C                    16                                    ⁢                                      X                    4                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    17                                    ⁢                                      Y                    5                                                  +                                                      C                    18                                    ⁢                                      Y                    4                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    19                                    ⁢                                      Y                    3                                    ⁢                                      X                    2                                                  +                                                      C                    20                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    21                                    ⁢                                      YX                    4                                                  +                                                      C                    22                                    ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    23                                    ⁢                                      Y                    6                                                  +                                                      C                    24                                    ⁢                                      Y                    5                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    25                                    ⁢                                      Y                    4                                    ⁢                                      X                    2                                                  +                                                      C                    26                                    ⁢                                      Y                    3                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    27                                    ⁢                                      Y                    2                                    ⁢                                      X                    4                                                  +                                                      C                    28                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                      C                    29                                    ⁢                                      X                    6                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    30                                    ⁢                                      Y                    7                                                  +                                                      C                    31                                    ⁢                                      Y                    6                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    32                                    ⁢                                      Y                    5                                    ⁢                                      X                    2                                                  +                                                      C                    33                                    ⁢                                      Y                    4                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                        xe2x80x83                            ⁢                                                                    C                    34                                    ⁢                                      Y                    3                                    ⁢                                      X                    4                                                  +                                                      C                    35                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                      C                    36                                    ⁢                                      YX                    6                                                  +                                                      C                    37                                    ⁢                                      "LeftBracketingBar"                                          X                      7                                        "RightBracketingBar"                                                                                                          (        c        )            
It should be noted that an anamorphic surface or a toric surface is also usable as a rotationally asymmetric surface.
The following is a description of conditions to be satisfied when an optical system is formed by using a rotationally asymmetric surface. In the variable focal length optical element and optical system according to the present invention, powers Px and Py are defined in the same way as in the paragraph [0049] in JP(A) 11-194267 with respect to light rays in two directions in YZ- and XZ-planes containing the optical axis and perpendicularly intersecting each other. That is, the axial principal ray and a light ray that is parallel to and at a slight distance away from the axial principal ray are made to enter the optical element or the optical system to define powers Px and Py, and the reciprocals of the powers are defined as focal lengths Fx and Fy. It is important that the ratio Fy/Fx should satisfy the following condition:
0.5 less than |Fy/Fx| less than 2xe2x80x83xe2x80x83(5) 
If |FY/Fx| is not larger than the lower limit of the condition (5), i.e. 0.5, the focal length in the X-direction becomes excessively long in comparison to the focal length in the Y-direction. Consequently, the image is formed excessively large in the X-direction. That is, a distorted image having a horizontally elongated rectangular shape is formed undesirably. If |Fy/Fx| is not smaller than the upper limit, i.e. 2, the focal length in the X-direction becomes excessively short in comparison to the focal length in the Y-direction. Consequently, the image becomes relatively small in the X-direction. That is, a distorted image having a vertically elongated rectangular shape is formed undesirably.
In the case of an optical system allowing the focal length to be changed as in the present invention, it is important that the above condition (5) should be satisfied in each of positions changed from one to another. If the condition (5) is not satisfied in all positions from the wide-angle end to the telephoto end or in all positions during focusing from an infinite distance to a near point, the change of image distortion during zooming or focusing becomes unnatural. Consequently, the image obtained becomes very difficult to see.
It is preferable to satisfy the following condition:
0.7 less than |Fy/Fx| less than 1.3xe2x80x83xe2x80x83(5-1) 
The meaning of the upper and lower limits of the condition (5-1) is the same as that of the upper and lower limits of the condition (5).
In Examples 1 to 11 (described later), Fx, Fy and Fy/Fx are as follows (W is the wide-angle end; S is the standard position; T is the telephoto end; D is the diopter; and the units of length are millimeter):
The above-described optical element and optical system according to the present invention can be used as an image pickup optical system, a projection optical system, an ocular optical system, etc.
Still other objects and advantages of the invention will in part be obvious and will in part be apparent from the specification.
The invention accordingly comprises the features of construction, combinations of elements, and arrangement of parts which will be exemplified in the construction hereinafter set forth, and the scope of the invention will be indicated in the claims.